Definition:Embedding (Topology)

This page is about embedding in the context of topology. For other uses, see embedding.

Definition

Let $A, B$ be topological spaces.

Let $f: A \to B$ be a mapping.

Let the image of $f$ be given the subspace topology.

Let the restriction $f {\restriction_{A \times f\sqbrk A }}$ of $f$ to its image be a homeomorphism.

Then $f$ is an embedding (of $A$ into $B$).

Also known as

An embedding is also known as a homeomorphism on its image.

Some authors use the terms topological embedding, imbedding, or topological imbedding.

Some refer to an embedding in rather than an embedding into.

Also defined as

Some authors define an embedding as a continuous injection.

Also see

• Continuous Injection from Compact Space to Hausdorff Space is Embedding

• Results about embeddings in the context of topology can be found here.

Sources

• 1955: John L. Kelley: General Topology: Chapter $4$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): embedding
• 2000: James R. Munkres: Topology (2nd ed.): $\S 18$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): embedding
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings